Fundamentally,
redox reactions are a family of reactions that are concerned with the
transfer of electrons between species. The term comes from the two
concepts of reduction and oxidation. It can be explained in simple terms:

Reduction is the gain of electrons or a decrease in oxidation state by a molecule, atom, or ion.

Although
oxidation reactions are commonly associated with the formation of
oxides from oxygen molecules, these are only specific examples of a more
general concept of reactions involving electron transfer.

Redox reactions, or oxidation-reduction reactions, have a number of similarities to acid-base reactions. Like acid-base reactions,
redox reactions are a matched set, that is, there cannot be an
oxidation reaction without a reduction reaction happening
simultaneously. The oxidation alone and the reduction alone are each
called a half-reaction,
because two half-reactions always occur together to form a whole
reaction. When writing half-reactions, the gained or lost electrons are
typically included explicitly in order that the half-reaction be balanced with respect to electric charge.

Though sufficient for many purposes, these descriptions are not precisely correct. Oxidation and reduction properly refer to a change in oxidation state — the actual transfer of electrons may never occur. Thus, oxidation is better defined as an increase in oxidation state, and reduction as a decrease in oxidation state.
In practice, the transfer of electrons will always cause a change in
oxidation state, but there are many reactions that are classed as
"redox" even though no electron transfer occurs (such as those involving covalent bonds).